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\input lib/e-macros
\input lib/macros
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\title{Parameter estimation for state-space models\\
through particle/MCMC filter}
\author{Fabien Campillo\thanks{\protect\url{Fabien.Campillo@inria.fr} --- 
           Project--Team MODEMIC, INRIA/INRA, UMR MISTEA, b\^at. 29, 2 place Viala, 34060 Montpellier cedex 06, France.}
           \and
        Vivien Rossi
      }
\date{12/12/2012\\[0.5em] {\small Dernière compilation: \today}}
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\newcommand{\bftheta}{\mathbf\theta}

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\section{Parameter estimation in state space models}
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We consider a hidden Markov model $(X_{1:T},Y_{1:T})$, with notation $X_{1:T}=(X_{1},\dots,X_{T})$, where the state process $X_{t}$ takes values in $\R^n$ and the observation process takes values in $\R^d$. We suppose that the model depend on a parameter $\theta$ which takes values in $\R^p$. Let $\rho(\rmd \theta)$ be the prior distribution on $\Theta$, $\nu^\theta(\rmd x_{1})$ be the initial distribution of $X_{1}$ conditionally on $\Theta=\theta$. The transition kernel of the Markov process $X_{t}$ conditionally on $\Theta=\theta$ is denoted:
\begin{subequations}
\label{eq.model}
\begin{align}
\label{eq.state}
   Q_{t}^\theta(x,\rmd x')
   &=
   \P(X_{t}\in\rmd x'|X_{t-1}=x,\Theta=\theta)\,.
\end{align}
We suppose that conditionally on $\Theta=\theta$ and on $X_{1:T}=x_{1:T}$ the observations $Y_{1},\dots,Y_{T}$ are independent and that the conditional distribution of $Y_{t}$ admits a density with respect to the Lebesgue measure, leading to the following likelihood:
\begin{align}
\label{eq.obs}
   g_t^{\theta}(y|x)
   &=
   \textrm{p.d.f. of }Y_{t}\textrm{  given }X_{t}=x\,,\ \Theta=\theta\,.
\end{align}
\end{subequations}
In case of missing data at instant $t$, $g_t^{\theta}(y|x)$ is constant and $y$ is a dummy observation.

\medskip

Hence the joint distribution of the $X_{1:K},Y_{1:K},\Theta$ is:
\begin{align*}
  \P(X_{1:T}\in\rmd x_{1:T},Y_{1:T}\in\rmd y_{1:T},\Theta\in\rmd \theta)
  &=
  \rho(\rmd \bftheta)\;
  \QQ^\theta(\rmd x_{1:T})\,\GG^\theta(y_{1:T}|x_{1:T})  \,\rmd y_{1:T}
\end{align*}
where:
\begin{align*}
  \QQ^\theta(\rmd x_{1:T})
  &\eqdef
  \nu^\theta(\rmd x_{1})\;\prod_{t=2}^T Q_{t}^\theta(x_{t-1},\rmd x_{t})
\end{align*}
is the conditional distribution of $X_{1:T}$ given $\Theta=\theta$, and
\begin{align*}
  \GG^\theta(y_{1:T}|x_{1:T})\;\rmd y_{1:T}
  &\eqdef
  \prod_{t=1}^T g_t^{\theta}(y_{t}|x_{t})\,\rmd y_{t}\,.
\end{align*}
is the conditional distribution of $Y_{1:T}$ given $X_{1:T}$ and $\Theta=\theta$.


%---------------------------------------------------------------
\begin{examples}
Consider the following nonlinear state-space model:
\begin{subequations}
\label{eq.generique}
\begin{align}
\label{eq.generique.state}
  X_{t+1}&=f_{\theta}(t,X_{t},W_{t})\,,\\
\label{eq.generique.obs}
  Y_{t}&=h_{\theta}(t,X_{t}) + \sigma_{\theta}(t,X_{t})\,V_{t}\,.
\end{align}
\end{subequations}
where, conditionally on $\Theta=\theta$, $W_{1:T}$, $V_{1:T}$ and $X_{1}$ are independent, and where $V_{1:T}$ is i.i.d. with density $p_{V}$. We suppose that $Y_{t}$ and $VV_{t}$ have the same dimension and that the matrix $\sigma_{\theta}(t,x)$ is regular.

The state-space model \eqref{eq.generique} is a special case of model \eqref{eq.model}. Note that:
\begin{align*}
  g_t^{\theta}(y|x)
  =
  \frac{1}{\det(\sigma_{\theta}(t,x))}\,p_{V}\Big(\sigma_{\theta}(t,x)^{-1}\,[y-h_{\theta}(t,x)]\Big)
\end{align*}
For example we may consider the following examples:
\begin{enumerate}



%--------------------
\item Linear-Gaussian case:
\begin{subequations}
\label{eq.ex1}
\begin{align}
\label{eq.ex1.state}
  X_{t+1}&=\alpha\,X_{t}+\sigma_{W}\,W_{t}\,,\\
\label{eq.ex1.obs}
  Y_{t}&=X_{t} + \sigma_{V}\,V_{t}
\end{align}
\end{subequations}
where $W_{t}\sim N(0,I)$, $V_{t}\sim N(0,I)$,
$X_{1}\sim N(\mu_{1},P_{1})$ are independent. 
Here $\theta=(\alpha,\sigma_{W},\sigma_{V})$ and
\begin{align*}
  Q^\theta_{t}(x,\,\cdot\,)
  &=
  N(\alpha\,x,\sigma_{W}^2)\,,
\\
  g_t^{\theta}(y|x)
  &=
  \frac{1}{\sqrt{2\,\pi\,\sigma_{V}^2}}\,
     \exp\Big(-\frac{|y-x|^2}{2\,\sigma_{V}^{2}}\Big)\,.
\end{align*}

 

%--------------------
\item stochastic volatility model \citep{pitt1999b}:
\begin{subequations}
\label{eq.ex2}
\begin{align}
\label{eq.ex2.state}
  X_{t+1}&=\alpha\,X_{t}+\sigma_{W}\,W_{t}\,,\\
\label{eq.ex2.obs}
  Y_{t}&=\beta\,\exp(X_{t}/2)\,V_{t}
\end{align}
\end{subequations}
where $W_{t}\sim N(0,I)$, $V_{t}\sim LN(0,I)$ (log-normal),
$X_{1}\sim N(\mu_{1},P_1)$ are independent.  
Here $\theta=(\alpha,\beta,\sigma_{W})$ and
\begin{align*}
  Q^\theta_{t}(x,\,\cdot\,)
  &=
  N(\alpha\,x,\sigma_{W}^2)\,,
\\
  g_t^{\theta}(y|x)
  &=
  \frac{1}{\sqrt{2\,\pi\,\sigma_{V}^2}}\,
     \exp\Big(-\frac{|y-x|^2}{2\,\sigma_{V}^{2}}\Big)\,.
\end{align*}


$X\sim LN(\mu,\sigma^2)$
\[
  f(x)
  =
  \frac{1}{\sqrt{2\,\pi\,\sigma^2}\times x}\,\exp\Big(-\frac{|\log(x)-\mu|^2}{2\,\sigma^2}\Big)
\]

%--------------------
\item  Ricker model \citep{meng-gao2012a}:
\begin{subequations}
\label{eq.ex3}
\begin{align}
\label{eq.ex3.state}
  X_{t+1}&=X_{t}+\alpha-\beta\,e^{X_{t}}+\sigma_{W}\,W_{t}\,,\\
\label{eq.ex3.obs}
  Y_{t}&=X_{t} + \sigma_{V}\,V_{t}\,.
\end{align}
Here $\theta=(\alpha,\beta,\sigma_{W},\sigma_{V})$ and
\begin{align*}
  Q^\theta_{t}(x,\,\cdot\,)
  &=
  N(x+\alpha-\beta\,e^x,\sigma_{W}^2)\,,
\\
  g_t^{\theta}(y|x)
  &=
  \frac{1}{\sqrt{2\,\pi\,\sigma_{V}^2}}\,
      \exp\Big(-\frac{|y-h_{\theta}(x)|^2}{2\,\sigma_{V}^{2}}\Big)\,.
\end{align*}
\end{subequations}
\end{enumerate}

\end{examples}
%---------------------------------------------------------------

\bigskip

The likelihood function for $\theta$ is:
\begin{align*}
  L(\theta)
  &=
  \P(Y_{1:T}\in\rmd y_{1:T}|\Theta=\theta)/\big(\rmd y_{1}\cdots\rmd y_{T}\big) 
  \\
  &
  = 
  \int_{x_{1}\in\R^n}\cdots \int_{x_{T}\in\R^n}
     \QQ^\theta(\rmd x_{1:T})\,\GG^\theta(y_{1:T}|x_{1:T})
  \\
  &
  = 
  \E\Big[\GG^\theta(y_{1:T}|X_{1:T})\Big|Y_{1:T}=y_{1:T},\Theta=\theta\Big]
  \vphantom{\int}\,.
\end{align*}


\section{A particle filter within EM algorithm for off line parameter estimation in space-state models}

Let $y_1,\dots,y_T$ the observations, $f$ the dynamic model for the state, $\psi_t$ the density function of the state $x_t$ conditionnaly to $x_{t-1}$ and $\theta$, $\phi_t$ the likelihood function of the state $x_t$ conditionnaly to  its observation $y_t$ and $\theta$. Let $\pi_\theta^0$, the prior distribution for the unknown parameters, $N$ the number of particle used in the interacted particle filter (IPF).

Algorithm : 
\begin{itemize}
\item[step 1] sample the parameter initial value $\theta^*\sim \pi_\theta^0$
\item[step 2] sample the noise values for the state-model : $w_i^j\sim \pi_Q$  with $i=1,\dots,N$, $j=1,\dots,T$.
\item[step 3] sample the random values for the resampling step of the particle filter : $u_i^j\sim\mathcal{U}[0,1]$ with $i=1,\dots,N$, $j=1,\dots,T$.
\item[step 4] compute $\hat x_t$, for $t=1,\dots,T$ the estimate of the expectation of the state at the time $t$, obtained by the IPF using $\theta^*$, $w_i^j$ and $u_i^j$ sampled at steps 2 and 3.
\item[step 5] maximize the completed likelihood function $\prod_{t=1}^T \phi_t(\hat x_t|y_t,\theta)\psi_t(\hat x_t|\hat x_{t-1},\theta)$ over $\theta$, then update $\theta^*$
\item[] repeat steps 4 and 5 until the values of the completed likelihood function stabilize
\end{itemize}

It is necessary to repaet several this algoritm to reduce the probability of reaching a local maximum. \\

Remarques : 
\begin{itemize}
\item il faudrait peut être utiliser un lisseur plutôt qu'un filtre
\item je pense que doit pouvoir prouver la convergence vers l'estimateur du maximum de vraisemblance lorsque $N$ tend vers l'infini en faisant simplement une inégalité triangulaire
\end{itemize}


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\section{References}
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\citep{poyiadjis2011a} to read

\citep{kantas2009a} to read

\citep{olsson2008a} bof ???

\citep{devalpine2002a} to read 

\citep{calder2003a} bof

\citep{meng-gao2012a} c'est de la daube mais bon, c'est exactement le sujet: ils utilise Particle MCMC \citep{andrieu2010a} comme \citep{golightly2011a}, \citep{rasmussen2011a} 

EM \citep{andrieu2003c}, \cite{schon2006a},  \cite{schon2011b}

froze randomness \cite{geyer1996a} \cite{hurzeler2001a}


à lire:  \cite{schon2011b}

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